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Partial Differential Equations with Random Input Data: A Perturbation Approach

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Abstract

We study the numerical approximation of partial differential equations with random input data. Such problems arise when the uncertainty of the underlying system is taken into account using a probability setting. The main goal of this paper is to review the perturbation approach used in [1] for the random space approximation. The idea of this technique is to expand the exact random solution in power series of a (small) parameter \(\varepsilon\) that characterizes the amount of uncertainty of the problem. This method yields deterministic problems that are decoupled for the coefficients building the term of a fixed power of \(\varepsilon\). Each problem can then be solved approximately using standard methods, such as the finite element method for the physical space discretization and an Euler scheme for time integration, as considered here. We apply the proposed methodology to several different problems, starting with an elliptic model problem with a random coefficient to facilitate the presentation. For each problem, focus is made on the derivation of (residual-based) a posteriori error estimates that take the various sources of error into account.

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Notes

  1. If the factor \(\frac{1}{2}\) is replaced by \(\frac{1}{4}\) for the jump contribution, see Remark 8, then we should take \(C_{H_0^1}:= 1/5\). See Appendix 7.2 for more details.

  2. Notice that we get comparable results if we bound these two terms separately, in which case the estimator due to the uncertainty reads \(\eta _{\varepsilon }^2=2\varepsilon ^2C_F^2\left( \Vert 2s_0f_0+s_0^2f_1\Vert _{L^2(D)}^2+b^2\Vert u_{0,h}u_{0,h}'\Vert _{L^2(D)}^2\right)\).

  3. This first example is similar to the case (3a) considered in [93]. The difference is that here we impose Robin (random) boundary conditions on a part of the boundary.

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Acknowledgements

I would like to express my deep appreciation to Prof. Fabio Nobile and Prof. Marco Picasso, my two PhD advisers, without whom this work would not have been possible. I thank them for their guidance, insight and constant support.

Funding

This study was partially funded the Swiss National Science Foundation (Grant No. P2ELP2-175056).

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Correspondence to Diane Guignard.

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Supported by the Swiss National Science Foundation, Grant P2ELP2-175056.

Appendices

Appendix for Section 2

1.1 Derivation of Problems (20), (21) and (22) and Link to the Derivatives

We give here more details about the derivation of problems (20), (21) and (22) that we need to solve to build the approximate solution \(u_0+\varepsilon u_1+\varepsilon ^2u_2\). In particular, we will see that the deterministic problems for the terms \(u_0\) and \(u_1\) are uniquely determined while those for \(u_2\) are not. We thus discuss the various ways to build the term \(u_2\). Moreover, we will make a more precise link between each term and the derivatives of \(u=u({\mathbf {x}},{\mathbf {y}})\) with respect to the \(y_j\), \(j=1,\ldots ,L\).

Recall that we assume that the diffusion coefficient a has the form

$$\begin{aligned} a({\mathbf {x}},\omega )=a({\mathbf {x}},{\mathbf {Y}}(\omega ))=a_0({\mathbf {x}})+\varepsilon \sum _{j=1}^La_j({\mathbf {x}})Y_j(\omega ). \end{aligned}$$

Moreover, the random solution u is expanded as

$$\begin{aligned} u({\mathbf {x}},{\mathbf {Y}}(\omega ))=u_0({\mathbf {x}})+\varepsilon u_1({\mathbf {x}},{\mathbf {Y}}(\omega ))+\varepsilon ^2 u_2({\mathbf {x}},{\mathbf {Y}}(\omega ))+\cdots \end{aligned}$$

with \(u_1=\sum _{j=1}^LU_jY_j\) and \(u_2=\sum _{j,k=1}^LU_{jk}Y_jY_k\). Similar expansion can be used for the higher order terms, see (69) where the general case is treated or [9, 94]. If we substitute the expansions of a and u in the first equation of problem (13), we get

$$\begin{aligned}&-\nabla \cdot \left( \left( a_0+\varepsilon \sum _{j=1}^La_jY_j\right) \nabla \left( u_0+\varepsilon \sum _{j=1}^LU_jY_j\right. \right. \\&\quad \left. \left. +\,\varepsilon ^2\sum _{j,k=1}^LU_{jk}Y_jY_k+\cdots \right) \right) =f. \end{aligned}$$

After recalling that f is deterministic by assumption, we separate then the terms of different order in \(\varepsilon\). The equation for the \({\mathcal {O}}(1)\) term is

$$\begin{aligned} -\nabla \cdot (a_0\nabla u_0)=f \end{aligned}$$

which yields problem (20) after adding suitable boundary conditions. Next, the equation for the \({\mathcal {O}}(\varepsilon )\) term is

$$\begin{aligned} -\varepsilon \sum _{j=1}^LY_j\nabla \cdot (a_0\nabla U_j+a_j\nabla u_0)=0. \end{aligned}$$
(214)

Since the set \(\{Y_j: \, j=1,\ldots ,L\}\) is orthonormal, it is in particular linearly independent. Therefore, Eq. (214) holds if and only if each term is zero, i.e.

$$\begin{aligned} \nabla \cdot (a_0\nabla U_j+a_j\nabla u_0)=0 \quad \forall j=1,\ldots ,L, \end{aligned}$$
(215)

which is nothing else than the first equation of problem (21). Notice that the relation (215) can also be obtained by multiplying (214) by \(Y_k\) and taking the ensemble mean, see [9], thanks again to the fact that \({\mathbb {E}}[Y_jY_k]=\delta _{jk}\). Finally, we collect the terms in \({\mathcal {O}}(\varepsilon ^2)\) to obtain

$$\begin{aligned} -\varepsilon ^2\sum _{j,k=1}^LY_jY_k\nabla \cdot (a_0\nabla U_{jk}+a_j\nabla U_k)=0. \end{aligned}$$
(216)

A sufficient condition for (216) to hold is that

$$\begin{aligned} \nabla \cdot (a_0\nabla U_{jk}+a_j\nabla U_k)=0 \quad \forall j,k=1,\ldots ,L, \end{aligned}$$
(217)

which corresponds to the set of PDEs in (22). However, it is not necessary to have (217) to verify (216) since the set \(\{Y_jY_k:\, j,k=1,\ldots ,L\}\) is not linearly independent. Using the fact that \(Y_jY_k=Y_kY_j\), we can rewrite (216) as

$$\begin{aligned}&-\varepsilon ^2\sum _{1\le j\le k\le L}Y_jY_k\nabla \cdot \left[ a_0\nabla (U_{jk}+U_{kj})+a_j\nabla U_k\right. \nonumber \\&\quad \left. +\,a_k\nabla U_j\right] \beta _{jk}=0 \end{aligned}$$
(218)

where \(\beta _{jk}=1-\frac{1}{2}\delta _{jk}\) is introduced to allow to keep the cases \(j<k\) and \(j=k\) under the same summation sign. Now, the set \(\{Y_jY_k:\, 1\le j\le k\le L\}\) is linearly independent [9] and thus (218) holds if and only if

$$\begin{aligned} \nabla \cdot (a_0\nabla (U_{jk}+U_{kj})+a_j\nabla U_k+a_k\nabla U_j)=0 \end{aligned}$$

for all \(1\le j\le k\le L\). If we write \(\tilde{U}_{jk}:=\frac{U_{jk}+U_{kj}}{2}\) for \(j,k=1,\ldots ,L\) we have then

$$\begin{aligned} u_2= & {} \sum _{j,k=1}^LU_{jk}Y_jY_k=\sum _{1\le j\le k\le L}\beta _{jk}(U_{jk}+U_{kj})Y_jY_k \\= & {} \sum _{j,k=1}^L\tilde{U}_{jk}Y_jY_k. \end{aligned}$$

Notice that \(\tilde{U}_{jk}\) solves

$$\begin{aligned} -\nabla \cdot \left( a_0\nabla \tilde{U}_{jk}+\frac{a_j\nabla U_k+a_k\nabla U_j}{2}\right) =0 \end{aligned}$$

for all \(j,k=1,\ldots ,L\) and that \(\tilde{U}_{jk}+\tilde{U}_{kj}=U_{jk}+U_{kj}\). The advantage of building \(u_2\) with the \(\tilde{U}_{jk}\) instead of the \(U_{jk}\) relies in the fact that \(\tilde{U}_{jk}=\tilde{U}_{kj}\) while \(U_{jk}\) is not necessarily equal to \(U_{kj}\). Therefore, the construction of \(u_2\) with the \(\tilde{U}_{jk}\) requires the resolution of \(\frac{L(L+1)}{2}\) problems whereas \(L^2\) problems need to be solved when the \(U_{jk}\) are used.

Notice that the problems we obtain for \(u_0\), \(U_j\), \(U_{jk}\) and \(U_{j_1j_2\cdots j_n}\), given by (20), (21), (22) and (69), respectively, are equivalent to those derived in [10]. In that paper, the authors apply what they called the method of successive approximations which uses the Karhunen–Loève expansion to represent the stochastic diffusion coefficient combined with the Neumann series expansion method. In fact, applied to the specific linear elliptic diffusion model problem (10), the (generalized or standard) Neumann expansion method and the perturbation method are equivalent [95].

In the remaining part of this section, we clarify the link between the various terms \(u_0\), \(U_j\), \(U_{jk}\) and \(\tilde{U}_{jk}\) defined above and the derivatives of \(u=u({\mathbf {x}},{\mathbf {y}})\) with respect to the \(y_j\). In other words, we compare the expansion (19) of u with its Taylor expansion around \({\mathbf {y}}_0={\mathbb {E}}[{\mathbf {Y}}]={\mathbf {0}}\). Recall that it has been proved (see for instance [48]) that the weak solution \(u=u({\mathbf {x}},{\mathbf {y}})\) of problem (13), i.e. the solution of (14), is analytic with respect to each variable \(y_j\), \(j=1,\ldots ,L\). First of all, we have

$$\begin{aligned} a({\mathbf {x}},{\mathbf {y}}_0)=a_0({\mathbf {x}}), \quad \frac{\partial a}{\partial y_j}({\mathbf {x}},{\mathbf {y}}_0)=\varepsilon a_j({\mathbf {x}}) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 a}{\partial y_k\partial y_j}({\mathbf {x}},{\mathbf {y}}_0)=0 \end{aligned}$$

for all \(j,k=1,\ldots ,L\). Then, we recall that for each \({\mathbf {y}}\in \varGamma\) the solution \(u(\cdot ,{\mathbf {y}})\in H_0^1(D)\) of problem (14) satisfies

$$\begin{aligned} \int _Da({\mathbf {x}},{\mathbf {y}})\nabla u({\mathbf {x}},{\mathbf {y}})\cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=\int _Df({\mathbf {x}})v({\mathbf {x}})d{\mathbf {x}}\end{aligned}$$
(219)

for all \(v\in H_0^1(D)\), \(\rho \text{-a.e. } {{\text{in }}} \varGamma\). The evaluation of Eq. (219) at \({\mathbf {y}}_0\) yields

$$\begin{aligned} \int _Da_0({\mathbf {x}})\nabla u({\mathbf {x}},{\mathbf {y}}_0)\cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=\int _Df({\mathbf {x}})v({\mathbf {x}})d{\mathbf {x}}. \end{aligned}$$
(220)

We can formally differentiate Eq. (219) with respect to \(y_j\) to get

$$\begin{aligned} \int _D\left( \frac{\partial a}{\partial y_j}\nabla u+a\nabla \frac{\partial u}{\partial y_j}\right) ({\mathbf {x}},{\mathbf {y}})\cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=0 \end{aligned}$$
(221)

for \(j=1,\ldots ,L\), and thus for \({\mathbf {y}}={\mathbf {y}}_0\) we have

$$\begin{aligned} \int _D\left( \varepsilon a_j({\mathbf {x}})\nabla u({\mathbf {x}},{\mathbf {y}}_0)+a_0({\mathbf {x}})\nabla \frac{\partial u}{\partial y_j}({\mathbf {x}},{\mathbf {y}}_0)\right) \cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=0. \end{aligned}$$
(222)

Taking then the derivative of (222) with respect to \(y_k\), or equivalently the second derivative of (219), we obtain for \(j,k=1,\ldots ,L\) the relation

$$\begin{aligned}&\int _D\left( \frac{\partial ^2 a}{\partial y_k\partial y_j}\nabla u+\frac{\partial a}{\partial y_j}\nabla \frac{\partial u}{\partial y_k}+\frac{\partial a}{\partial y_k}\nabla \frac{\partial u}{\partial y_j}\right. \\&\quad \left. +\,a\nabla \frac{\partial ^2 u}{\partial y_k\partial y_j}\right) ({\mathbf {x}},{\mathbf {y}})\cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=0. \end{aligned}$$

Since \(\frac{\partial ^2 a}{\partial y_k\partial y_j}=0\), the evaluation of last relation at \({\mathbf {y}}_0\) gives us

$$\begin{aligned}&\int _D\left( \varepsilon a_j({\mathbf {x}})\nabla \frac{\partial u}{\partial y_k}({\mathbf {x}},{\mathbf {y}}_0)+\varepsilon a_k({\mathbf {x}})\nabla \frac{\partial u}{\partial y_j}({\mathbf {x}},{\mathbf {y}}_0)\right. \nonumber \\&\quad \left. +\,a_0({\mathbf {x}})\nabla \frac{\partial ^2 u}{\partial y_k\partial y_j}({\mathbf {x}},{\mathbf {y}}_0)\right) \cdot \nabla v({\mathbf {x}})d{\mathbf {x}}=0 \end{aligned}$$
(223)

for \(j,k=1,\ldots ,L\). Finally, based on Eqs. (220), (222) and (223) we conclude that

$$\begin{aligned} u_0=u(\cdot ,{\mathbf {y}}_0), \quad \varepsilon ^2(U_{jk}+U_{kj})=\frac{\partial ^2 u}{\partial y_k\partial y_j}(\cdot ,{\mathbf {y}}_0) \end{aligned}$$

and

$$\begin{aligned} \varepsilon U_j=\frac{\partial u}{\partial y_j}(\cdot ,{\mathbf {y}}_0), \quad \varepsilon ^2\tilde{U}_{jk}=\frac{1}{2}\frac{\partial ^2 u}{\partial y_k\partial y_j}(\cdot ,{\mathbf {y}}_0) \end{aligned}$$

for \(j,k=1,\ldots ,L\).

1.2 Estimation of the Interpolation Constant

In this section, we briefly present the value of the interpolation constant \(C_{H_0^1}\) that can be included in the error estimator to get a sharp spatial error estimator. This value, which depends on the degree of the finite element space as well as if we are in 1D, 2D or 3D, can be estimated as follows: consider the (deterministic) Poisson problem \(-\varDelta u_0=f\) with homogeneous Dirichlet boundary conditions for which the exact solution is known. Define then \(C_{H_0^1}:=\Vert u_0-u_{0,h}\Vert _{H_0^1(D)}/\eta _0\) for h small enough, recalling that \(\eta _0\) is the part of estimator controlling the FE approximation of \(u_0\). Notice that this estimation can be done once for all since \(C_{H_0^1}\) does not depend on the input data.

In the one-dimensional case, we have already mentioned that the constant for \({\mathbb {P}}_1\) finite element can be set to \(C_{H_0^1}=\frac{1}{3.46}\approx \frac{1}{2\sqrt{3}}\). The latter corresponds to the theoretical value \(\left( \frac{1}{p+1}\right) ^{1/p}\frac{1}{2}\) with \(p=2\) given in [96].

For the 2D case, let us consider \(D=(0,1)^2\) and \(u_0(x_1,x_2)=\sin (2\pi x_1)\sin (4\pi x_2)\) the solution of the Poisson problem with right-hand-side

$$\begin{aligned} f(x_1,x_2)=20\pi ^2\sin (2\pi x_2)\sin (4\pi x_2). \end{aligned}$$
(224)

We give in Table 19 the error \(\Vert \nabla (u_0-u_{0,h})\Vert _{L^2(D)}\) and the two estimators \(\eta _0\) and \(\hat{\eta }_0\) defined by

$$\begin{aligned} \eta _0^2:=\sum _{K\in {\mathcal {T}}_h}h_K^2\Vert f+\varDelta u_h\Vert _{L^2(K)}^2+\sum _{e\in {\mathcal {E}}_h}h_e\Vert [\nabla u_h\cdot \mathbf {n_e}]_{\mathbf {n_e}}\Vert _{L^2(e)}^2 \end{aligned}$$

and

$$\begin{aligned} \hat{\eta }_0^2:= & {} \sum _{K\in {\mathcal {T}}_h}\big [h_K^2\Vert f+\varDelta u_h\Vert _{L^2(K)}^2 \\&+\,\frac{1}{4}\sum _{e\subset \partial K}h_e\Vert [\nabla u_h\cdot \mathbf {n_e}]_{\mathbf {n_e}}\Vert _{L^2(e)}^2\big ]. \end{aligned}$$

We consider both structured and Delaunay triangulations with \(N=256\) equidistant vertices on each boundary of D, see Fig. 15 where the meshes for the case \(N=16\) are given.

Fig. 15
figure 15

Structured (left) and Delaunay (right) triangulations of D with \(N=16\)

The constant \(C_{H_1^0}\) can then be set to \(\Vert \nabla (u_0-u_{0,h})\Vert _{L^2(D)}/\eta _0\) or \(\Vert \nabla (u_0-u_{0,h})\Vert _{L^2(D)}/\hat{\eta }_0\) depending on the definition of the estimator.

Table 19 Error, estimator and effectivity index for the Poisson problem

Notice that we get similar values when considering other cases than (224). We see from the results of Table 19 that, as expected, the interpolation constant depends on the polynomial degree of the finite elements. Moreover, we could go further by estimating separately the efficiency of the interior residual and the contribution of the jump terms, but we will not do it here.

1.3 Upper and Lower Bounds for the Error \(u-u_{0,h}\) in the \(L_P^2(\varOmega ;H_0^1(D))\) Norm

We give here an a posteriori error estimator for the error \(\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}\) for which both upper and lower bounds can be shown. The spatial error estimator is the same, namely \(\eta _0\) given in (44). For the stochastic error, recall that we need to estimate

$$\begin{aligned} r(v;{\mathbf {y}}):= & {} {\mathcal {R}}(v;{\mathbf {y}})-{\mathcal {R}}(v;{\mathbf {y}}_0) \nonumber \\= & {} -\int _D(a(\cdot ,{\mathbf {y}})-a_0)\nabla u_{0,h}\cdot \nabla v, \end{aligned}$$
(225)

using \({\mathbf {y}}_0={\mathbb {E}}[{\mathbf {Y}}]={\mathbf {0}}\) and the definition in (42) of the residual \({\mathcal {R}}\). Contrary to Sect. 2.5, where a natural estimation is performed using the Cauchy–Schwarz inequality, we get here a stochastic error estimator by computing (approximately) the dual norm of r. More precisely, we have \(\Vert r(\cdot ;{\mathbf {y}})\Vert _{H^{-1}(D)}=\Vert \nabla w(\cdot ,{\mathbf {y}})\Vert _{L^2(D)}\) with \(w(\cdot ,{\mathbf {y}})\) the solution of

$$\begin{aligned} \int _D\nabla w(\cdot ,{\mathbf {y}})\cdot \nabla v=r(v;{\mathbf {y}}) \quad \forall v\in H_0^1(D), \, \rho \text{-a.e. } {{\text{in }}} \varGamma . \end{aligned}$$
(226)

We write then \(w({\mathbf {x}},{\mathbf {Y}}(\omega ))=\varepsilon \sum _{j=1}^LW_j({\mathbf {x}})Y_j(\omega )\) with \(W_j\in H_0^1(D)\) such that

$$\begin{aligned} \int _D\nabla W_j\cdot \nabla v=-\int _Da_j\nabla u_{0,h}\cdot \nabla v \quad \forall v\in H_0^1(D). \end{aligned}$$

Let \(w_h({\mathbf {x}},{\mathbf {Y}}(\omega ))=\varepsilon \sum _{j=1}^LW_{j,h}({\mathbf {x}})Y_j(\omega )\), where \(W_{j,h}\in V_h\) is the FE approximation of \(W_j\). The error estimator can then be defined as

$$\begin{aligned} \hat{\eta }^2= \left( \eta _0^2+\hat{\eta }_1^2\right) ^{\frac{1}{2}} \quad \text{ with } \quad \hat{\eta }_1^2:=\varepsilon ^2\sum _{j=1}^L\Vert \nabla W_{j,h}\Vert _{L^2(D)}^2. \end{aligned}$$
(227)

To simplify the notation, let \(R\) and \(J\) denote the interior element and the jump residuals defined on each \(K\in {\mathcal {T}}_h\) and each \(e\in {\mathcal {E}}_h\) by respectively

$$\begin{aligned} {R\,}_{\vert _{\,K}}={(f+\nabla \cdot (a_0\nabla u_{0,h}))\,}_{\vert _{\,K}} \end{aligned}$$

and

$$\begin{aligned} {J\,}_{\vert _{\,e}}=\left[ a_0\nabla u_{0,h}\cdot {\mathbf {n}}_e\right] _{{\mathbf {n}}_e}. \end{aligned}$$

The error estimator \(\eta _0\) can then be written

$$\begin{aligned} \eta _0^2=\sum _{K\in {\mathcal {T}}_h}\eta _K^2, \,\eta _K^2:=h_K^2\Vert R\Vert _{L^2(K)}^2+\frac{1}{2}\sum _{e\subset \partial K}h_e\Vert J\Vert _{L^2(e)}^2 \end{aligned}$$

The goal here is to prove that the error estimator \(\hat{\eta }\) defined in (227) is equivalent to the error \(\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}\). We assume from now on that \(D\subset {\mathbb {R}}^d\) with \(d=2\), mentioning that the case \(d=1\) can be treated easily since no jump terms occur while the extension to the case \(d=3\) is straightforward.

To prove the spatial lower bound, see (235), we will need some definitions and notation that we introduce now. For any element \(K\in {\mathcal {T}}_h\), using the notation given in Fig. 16-left, we define the so-called element bubble function \(\psi _K\) and edge bubble function \(\psi _{e_i}\), see for instance [30], by

$$\begin{aligned} \psi _K=27\lambda _1\lambda _2\lambda _3 \quad \text{ and } \quad \psi _{e_i}=4\lambda _{i+1}\lambda _{i+2}, \quad i=1,2,3, \end{aligned}$$

where the indices are taken modulo 3 and \(\lambda _1,\lambda _2,\lambda _3\) are the (linear) barycentric coordinates on K. Using the notation used in [30], we denote by \(w_K\) the union of all the elements sharing an edge with K and, for an internal edge e, we write \(w_e\) the union of the two elements sharing e as an edge, see Fig. 16 for an illustration.

Fig. 16
figure 16

Notation for an element K in \({\mathcal {T}}_h\) (left) and illustration of the domains \(w_K\) (middle) and \(w_e\) (right)

The bubble functions satisfy the following properties: for any polynomial \(\varphi\) of degree less or equal to k we have

$$\begin{aligned} \Vert \varphi \Vert _{L^2(K)}\le & {} c_1\Vert \psi _K^{\frac{1}{2}}\varphi \Vert _{L^2(K)} \end{aligned}$$
(228)
$$\begin{aligned} \Vert \nabla (\psi _K\varphi )\Vert _{L^2(K)}\le & {} c_2h_K^{-1}\Vert \varphi \Vert _{L^2(K)} \end{aligned}$$
(229)
$$\begin{aligned} \Vert \varphi \Vert _{L^2(e)}\le & {} c_3\Vert \psi _e^{\frac{1}{2}}\varphi \Vert _{L^2(e)} \end{aligned}$$
(230)
$$\begin{aligned} \Vert \nabla (\psi _e\varphi )\Vert _{L^2(w_e)}\le & {} c_4h_e^{-\frac{1}{2}}\Vert \varphi \Vert _{L^2(e)} \end{aligned}$$
(231)
$$\begin{aligned} \Vert \psi _e\varphi \Vert _{L^2(w_e)}\le & {} c_5h_e^{\frac{1}{2}}\Vert \varphi \Vert _{L^2(e)} \end{aligned}$$
(232)

where the constants \(c_i\), \(i=1,\ldots ,5\), depend only on k and on the shape regularity parameter of \({\mathcal {T}}_h\) given in (23). Moreover, we have \(0\le \psi _K({\mathbf {x}})\le 1\) for all \({\mathbf {x}}\in K\), \(\psi _K({\mathbf {x}})=0\) for all \({\mathbf {x}}\notin K\), \(\max _{{\mathbf {x}}\in K}\psi _K({\mathbf {x}})=1\), \(0\le \psi _e({\mathbf {x}})\le 1\) for all \({\mathbf {x}}\in w_e\), \(\psi _e({\mathbf {x}})=0\) for all \({\mathbf {x}}\notin w_e\) and \(\max _{{\mathbf {x}}\in w_e}\psi _e({\mathbf {x}})=1\). For any element K, we denote by \(\bar{g}_K\) the mean value of g on K and similarly we denote by \(\bar{g}_e\) the mean value of g on any internal edge e, i.e.

$$\begin{aligned} \bar{g}_K=\frac{1}{|K|}\int _Kg \quad \text{ and } \quad \bar{g}_e=\frac{1}{|e|}\int _eg. \end{aligned}$$

Finally, we introduce the oscillation term\(\theta _K\) defined by

$$\begin{aligned} \theta _K^2:=\sum _{T\subset w_K}h_{T}^2\Vert R-\bar{R}_{T}\Vert _{L^2(T)}^2+\sum _{e\subset \partial K}h_e\Vert J-\bar{J}_e\Vert _{L^2(e)}^2. \end{aligned}$$
(233)

We can now state the upper and lower bounds, given in the following proposition.

Proposition 26

Letube the weak solution of problem (10) and let\(u_{0,h}\)be the solution of problem (36), respectively. There exist two constants\(C_0,C_1>0\)depending only on the mesh aspect ratio and\(s\in (0,1]\)such that

$$\begin{aligned}&\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}\le \frac{1}{a_{{{ min}}}}\left( C_0\eta _0+\hat{\eta }_1\right) +{\mathcal {O}}(\varepsilon h^s), \end{aligned}$$
(234)
$$\begin{aligned}&\quad \eta _0 \le C_1\big [a_{{{ max}}}\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}+\hat{\eta }_1 \nonumber \\&\qquad +\left( \sum _{K\in {\mathcal {T}}_h}\theta _K^2\right) ^{\frac{1}{2}}\big ] +{\mathcal {O}}(\varepsilon h^s) \end{aligned}$$
(235)

and

$$\begin{aligned} \hat{\eta }_1 \le a_{{{ max}}}\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}+C_0\eta _0+{\mathcal {O}}(\varepsilon h^s). \end{aligned}$$
(236)

Proof

We first derive a bound for the \(L_P^2(\varOmega ;H_0^1(D))\) norm of \(w\) (resp. \(w_h\)) in term of the norm of \(w_h\) (resp. \(w\)) and higher order terms, where \(w\) is the solution (226) and \(w_h\) its FE approximation. Let us introduce \(\phi ({\mathbf {x}},{\mathbf {Y}}(\omega ))=\varepsilon \sum _{j=1}^L\phi _j({\mathbf {x}})Y_j(\omega )\), where \(\phi _j\in H_0^1(D)\) is the solution of

$$\begin{aligned} \int _D\nabla \phi _j\cdot \nabla v = -\int _D a_j\nabla u_0\cdot \nabla v \quad \forall v\in H_0^1(D), \end{aligned}$$

and let \(\phi _h\) denotes its FE approximation. Notice that \(\phi (\cdot ,{\mathbf {Y}}(\omega ))\) solves

$$\begin{aligned} \int _D\nabla \phi \cdot \nabla v = -\int _D(a-a_0)\nabla u_0\cdot \nabla v \end{aligned}$$

for all \(v\in H_0^1(D)\), a.s. in \(\varOmega\), which is similar to the problem (226) for \(w\), except that \(u_{0,h}\) is replaced by \(u_0\) in the right-hand side. Thanks to the triangle inequality, we obtain

$$\begin{aligned} \Vert \nabla w\Vert _{L^2(D)}\le & {} \Vert \nabla w_h\Vert _{L^2(D)}+\Vert \nabla (w-\phi )\Vert _{L^2(D)} \\&+\Vert \nabla (\phi -\phi _h)\Vert _{L^2(D)}+\Vert \nabla (\phi _h-w_h)\Vert _{L^2(D)} \end{aligned}$$

from which we can deduce

$$\begin{aligned} \Vert \nabla w\Vert _{L_P^2(\varOmega ;L^2(D))} \le \Vert \nabla w_h\Vert _{L_P^2(\varOmega ;L^2(D))}+C\varepsilon h^s, \end{aligned}$$

where \(s\in (0,1]\) depends only on the regularity of \(u_0\), \(\phi _j\), \(j=1,\ldots ,L\), and the domain D [97, 98] and C is a (deterministic) positive constant independent of h and \(\varepsilon\) but dependent on the mesh aspect ratio, \(|u_0|_{H^{1+s}(D)}\) and \(|\phi _j|_{H^{1+s}(D)}\), \(j=1,\ldots ,L\). Therefore, recalling that \(w_h=\varepsilon \sum _{j=1}^LW_{j,h}Y_j\) and using \({\mathbb {E}}[Y_iY_j]=\delta _{ij}\) we get

$$\begin{aligned} \Vert \nabla w\Vert _{L_P^2(\varOmega ;L^2(D))} \le \hat{\eta }_1+C\varepsilon h^s \end{aligned}$$
(237)

with \(\hat{\eta }_1\) given in (227). Finally, proceeding in the same way we can obtain the relation

$$\begin{aligned} \hat{\eta }_1=\Vert \nabla w_h\Vert _{L_P^2(\varOmega ;L^2(D))} \le \Vert \nabla w\Vert _{L_P^2(\varOmega ;L^2(D))}+C\varepsilon h^s. \end{aligned}$$
(238)

We now prove the three bounds (234), (235) and (236) separately. The proof of (235) is inspired by what is done in [30, 99], while the idea for the proof of (236) is based on the proof of efficiency of the error estimator proposed in [100] for the Reduced Basis method. In the sequel, all the equations hold a.s. in \(\varOmega\) without specifically mentioning it.

Upper bound The proof is similar to the one of Proposition 3, only the bound of term controlling the stochastic error is different. For any \(v\in H_0^1(D)\), taking \(v_h={\mathcal {I}}_h v\) the Clément interpolant of \(v\) we have

$$\begin{aligned}&\int _Da\nabla (u-u_{0,h})\cdot \nabla v = \int _Dfv-\int _Da_0\nabla u_{0,h}\cdot \nabla v \\&\qquad -\int _D(a-a_0)\nabla u_{0,h}\cdot \nabla v \\&\quad = \int _Df(v-v_h)-\int _Da_0\nabla u_{0,h}\cdot \nabla (v-v_h)+\int _D\nabla w\cdot \nabla v \\&\quad \le \left[ C_0\left( \sum _{K\in {\mathcal {T}}_h}\eta _K^2\right) ^{\frac{1}{2}}+\Vert \nabla w\Vert _{L^2(D)}\right] \Vert \nabla v\Vert _{L^2(D)}, \end{aligned}$$

where \(C_0\) depends only on the constants in (26) and (28). Since \(a_{{{ min}}}\) is a lower bound for a, taking \(v=u-u_{0,h}\) we get

$$\begin{aligned} a_{{{ min}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(D)}\le C_0\eta _0+\Vert \nabla w\Vert _{L^2(D)} \end{aligned}$$

and thus, taking the \(L_P^2(\varOmega )\) norm on both sides of the last inequality we have

$$\begin{aligned} a_{{{ min}}}\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}\le C_0\eta _0+\Vert \nabla w\Vert _{L_P^2(\varOmega ;L^2(D))}. \end{aligned}$$

Finally, we obtain (234) using (237).

h-lower bound First of all, notice that for any \(v\in H_0^1(D)\) we have

$$\begin{aligned}&\int _Da\nabla (u-u_{0,h})\cdot \nabla v = \sum _{K\in {\mathcal {T}}_h}\int _KRv+\sum _{e\in {\mathcal {E}}_h}\int _eJv \nonumber \\&\qquad -\int _D(a-a_0)\nabla u_{0,h}\cdot \nabla v \nonumber \\&\quad =\sum _{K\in {\mathcal {T}}_h}\int _KRv+\sum _{e\in {\mathcal {E}}_h}\int _eJv+\int _D\nabla w\cdot \nabla v. \end{aligned}$$
(239)

The proof is then divided into three steps.

  1. 1.

    Let K be any element in \({\mathcal {T}}_h\) and let \(v_K=\bar{R}_K\psi _K\). We take \(v=v_K\) in (239). Since \(\text{ supp }\psi _K\subset K\), we have

    $$\begin{aligned} \int _Ka\nabla (u-u_{0,h})\cdot \nabla v_K= & {} \int _K\bar{R}_Kv_K+\int _K(R-\bar{R}_K)v_K \\&+\int _K\nabla w\cdot \nabla v_K \end{aligned}$$

    and thus, using the properties of the element bubble function given in (228) and (229), we obtain

    $$\begin{aligned} h_K\Vert \bar{R}_K\Vert _{L^2(K)}\le & {} c_1^2c_2a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(K)} \\&+\,c_1^2c_2\Vert \nabla w\Vert _{L^2(K)}+c_1^2h_K\Vert R-\bar{R}_K\Vert _{L^2(K)}. \end{aligned}$$

    Thanks to triangle’s inequality, we finally obtain

    $$\begin{aligned} h_K\Vert R\Vert _{L^2(K)}\le & {} c_1^2c_2a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(K)} \nonumber \\&+\,c_1^2c_2\Vert \nabla w\Vert _{L^2(K)} \nonumber \\&+\,(1+c_1^2)h_K\Vert R-\bar{R}_K\Vert _{L^2(K)}. \end{aligned}$$
    (240)
  2. 2.

    Let e be any interior edge of \({\mathcal {T}}_h\), let \(v_e=\bar{J}_e\psi _e\) and let \(K_1\) and \(K_2\) be the two elements that share e as an edge. We take \(v=v_e\) in (239) to get

    $$\begin{aligned} \int _{w_e}a\nabla (u-u_{0,h})\cdot \nabla v_e= & {} \sum _{K\in w_e}\int _KRv_e+\int _e\bar{J}_ev_e \\&\quad +\int _e(J-\bar{J}_e)v_e+\int _{w_e}\nabla w\cdot \nabla v_e. \end{aligned}$$

    Therefore, using the properties of the edge bubble function given in (230), (231) and (232), we obtain

    $$\begin{aligned} h_e^{\frac{1}{2}}\Vert \bar{J}_e\Vert _{L^2(e)}\le & {} c_3^2c_4a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(w_e)} \\&\quad +\,c_3^2c_5h_e\Vert R\Vert _{L^2(w_e)}+c_3^2h_e^{\frac{1}{2}}\Vert J-\bar{J}_e\Vert _{L^2(e)} \\&\quad +\,c_3^2c_4\Vert \nabla w\Vert _{L^2(w_e)} \end{aligned}$$

    and thus

    $$\begin{aligned} h_e^{\frac{1}{2}}\Vert J\Vert _{L^2(e)}\le & {} c_3^2c_4a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(w_e)} \\&+\,c_3^2c_5h_e\Vert R\Vert _{L^2(w_e)}+(1+c_3^2)h_e^{\frac{1}{2}}\Vert J-\bar{J}_e\Vert _{L^2(e)} \\&+\,c_3^2c_4\Vert \nabla w\Vert _{L^2(w_e)} \\\le & {} \sum _{i=1}^2\big [a_{{{ max}}}c_3^2(c_4+c_1^2c_2c_5)\Vert \nabla (u-u_{0,h})\Vert _{L^2(K_i)} \\&+\,(1+c_1^2)c_3^2c_5h_{K_i}\Vert R-\bar{R}_{K_i}\Vert _{L^2(K_i)} \\&+\,c_3^2(c_4+c_1^2c_2c_5)\Vert \nabla w\Vert _{L^2(K_i)}\big ] \\&+\,(1+c_3^2)h_e^{\frac{1}{2}}\Vert J-\bar{J}_e\Vert _{L^2(e)} \end{aligned}$$

    using relation (240).

  3. 3.

    Putting everything together, we obtain for any element \(K\in {\mathcal {T}}_h\)

    $$\begin{aligned} \eta _K^2= & {} h_K^2\Vert R\Vert _{L^2(K)}^2+\frac{1}{2}\sum _{e\subset \partial K}h_e\Vert J\Vert _{L^2(e)}^2 \\\le & {} C_1\left( a_{{{ max}}}^2\Vert \nabla (u-u_{0,h})\Vert _{L^2(w_K)}^2+\Vert \nabla w\Vert _{L^2(w_K)}^2\right. \\&\left. +\,\sum _{T\subset w_K}h_{T}^2\Vert R-\bar{R}_{T}\Vert _{L^2(T)}^2+\sum _{e\subset \partial K}h_e\Vert J-\bar{J}_e\Vert _{L^2(e)}^2\right) , \end{aligned}$$

    where \(C_1\) depends only on the regularity of the mesh (through the constants \(c_i\), \(i=1,\ldots ,5\)). Recalling the definition of \(\theta _K\) in (233), if we sum over all \(K\in {\mathcal {T}}_h\) and use the relation \((a^2+b^2+c^2)\le (a+b+c)^2\) valid for any non-negative numbers abc, we get

    $$\begin{aligned} \eta _0\le & {} C_1\left[ a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(D)}+\Vert \nabla w\Vert _{L^2(D)}\right. \\&\left. +\,\left( \sum _{K\in {\mathcal {T}}_h}\theta _K^2\right) ^{\frac{1}{2}}\right] \end{aligned}$$

    where \(C_1\) has changed but still only depends on the mesh aspect ratio. Finally, we obtain (235) taking the \(L_P^2(\varOmega )\) norm and using (237).

\(\varepsilon\)-lower bound For any \(v\in H_0^1(D)\) we have

$$\begin{aligned} \int _D\nabla w\cdot \nabla v= & {} -\int _D(a-a_0)\nabla u_{0,h}\cdot \nabla v \nonumber \\&=\int _Da\nabla (u-u_{0,h})\cdot \nabla v-\int _Da_0\nabla (u_0-u_{0,h})\cdot \nabla v. \end{aligned}$$
(241)

Taking \(v=w\) in (241) and noticing that the last term of (241) is nothing else than (minus) the residual for \(u_{0,h}\), we can easily derive the bound

$$\begin{aligned} \Vert \nabla w\Vert _{L^2(D)}^2\le & {} \left[ a_{{{ max}}}\Vert \nabla (u-u_{0,h})\Vert _{L^2(D)}\right. \\&\left. +\,C_0\left( \sum _{K\in {\mathcal {T}}_h}\eta _K^2\right) ^{\frac{1}{2}}\right] \Vert \nabla w\Vert _{L^2(D)} \end{aligned}$$

where \(C_0\) depends only on the constants in (26) and (28). From the last relation, we deduce taking the \(L_P^2(\varOmega )\) that

$$\begin{aligned} \Vert \nabla w\Vert _{L_P^2(\varOmega ;L^2(D))} \le a_{{{ max}}}\Vert u-u_{0,h}\Vert _{L_P^2(\varOmega ;H_0^1(D))}+C_0\eta _0 \end{aligned}$$

which conclude the proof thanks to (238). \(\square\)

Remark 29

Since \(u_{0,h}\) is piecewise affine, if \(a_0\) is piecewise constant then we have \(R=f\) and \(J=\bar{J}_e\). Therefore, in this case \(\theta^2 _K\) reduces to \(\sum _{T\subset w_K}h_{T}^2\Vert f-\bar{f}_{T}\Vert _{L^2(T)}^2\) which does no longer depend on \(u_{0,h}\). It is often refereed to as data oscillation.

Remark 30

We deduce from the three relations (234), (235) and (236) that

$$\begin{aligned} a_{{{ min}}}\le \frac{\hat{\eta }_1}{\Vert u-u_{0,h}\Vert }\le a_{{{ max}}}\quad \text{ as } h\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} C_0^{-1}a_{{{ min}}}\le \frac{\eta _0}{\Vert u-u_{0,h}\Vert }\le C_1a_{{{ max}}}\quad \text{ as } \varepsilon \rightarrow 0, \end{aligned}$$

where \(\Vert \cdot \Vert\) denotes the \(L^2(\varOmega ;H_0^1(D))\) norm and \(C_0\) and \(C_1\) are two positive constants depending only on the mesh aspect ratio.

Appendix for Section 4

1.1 Choice of the Norm

We motivate here the choice of the norm on the space \(V\times Q\) used in (173), more precisely regarding the scaling with the kinematic viscosity \(\nu\). Recalling that \({\Arrowvert}\,\, {\cdot}\,\, {\Arrowvert}\) denotes the \({L^{2}(D)}\) norm, we claim that the appropriate scaling is given by

$$\begin{aligned} {\left| \!\left| \!\left| {{\mathbf {v}},q}\right| \!\right| \!\right| }_k^2:=\nu ^k\Vert \nabla {\mathbf {v}}\Vert ^2+\nu ^{k-2}\Vert q\Vert ^2 \end{aligned}$$
(242)

for any choice \(k=0,1,2\). This choice is guided by the dimension unit of \(\nu\), p and \(\nabla {\mathbf {u}}\). Indeed, the dimension unit of the kinematic viscosity is \([\nu ]=\frac{m^2}{s}\) while we have, recall that p corresponds to the pressure divided by the density of the fluid,

$$\begin{aligned}{}[|\nabla {\mathbf {u}}|^2]=\left( \frac{1}{m}\cdot \frac{m}{s}\right) ^2=\frac{1}{s^2} \end{aligned}$$

and

$$\begin{aligned}{}[p^2]=\left( \frac{N}{m^2}\cdot \frac{m^3}{kg}\right) ^2=\frac{m^4}{s^4}, \end{aligned}$$

from which we deduce that \([\nu ^k|\nabla {\mathbf {u}}|]=[\nu ^{k-2}p^2]\) for all k. We mention that the scaling (242) is also the one that arises naturally when analyzing the a priori estimates on the solution or when performing the a posteriori error analysis. For simplicity, let us consider the (deterministic) Stokes problem given under the weak form by: find \(({\mathbf {u}},p)\in V\times Q\) such that

$$\begin{aligned} \left\{ \begin{array}{ll} a({\mathbf {u}},{\mathbf {v}})+b({\mathbf {v}},p) = F({\mathbf {v}}) & \quad \forall {\mathbf {v}}\in V\\ b({\mathbf {u}},q) = 0 & \quad \forall q\in Q, \end{array} \right. \end{aligned}$$

with \(V=[H_0^1(D)]^d\), \(Q=L_0^2(D)\), \(a({\mathbf {u}},{\mathbf {v}})=\nu \int _D\nabla {\mathbf {u}}:\nabla {\mathbf {v}}\), \(b({\mathbf {v}},q)=-\int _Dq\nabla \cdot {\mathbf {v}}\) and \(F({\mathbf {v}})=\int _D{\mathbf {f}}\cdot {\mathbf {v}}\). The bilinear form a is continuous and coercive on V with constant \(\nu\) and b is continuous on V with constant 1 and satisfies the inf-sup condition with constant \(\beta =\beta (D)\). The problem is thus well-posed (see [81]) and the following a priori estimates are satisfied

$$\begin{aligned} \Vert \nabla {\mathbf {u}}\Vert \le \frac{1}{\nu }\Vert F\Vert _{V'} \end{aligned}$$

and

$$\begin{aligned} \Vert p\Vert \le \frac{1}{\beta }\left( \Vert F\Vert _{V'}+\nu \Vert \nabla {\mathbf {u}}\Vert \right) \le \frac{2}{\beta }\Vert F\Vert _{V'}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \nu ^{k/2}\Vert \nabla {\mathbf {u}}\Vert +\nu ^{k/2-1}\Vert p\Vert \le C\nu ^{k/2-1}\Vert {\mathbf {f}}\Vert _{V'} \quad \forall k, \end{aligned}$$

where \(C=(1+2/\beta )\) is independent of \(\nu\), which is consistent with the scaling (242). Finally, for the a posteriori error analysis, denoting \(e={\mathbf {u}}-{\mathbf {u}}_h\) and \(E=p-p_h\) with \({\mathbf {u}}_h\) and \(p_h\) the finite element approximation of \({\mathbf {u}}\) and p, respectively, we have for any \(({\mathbf {v}},q)\in V\times Q\)

$$\begin{aligned} a({\mathbf {e}},{\mathbf {v}})+b({\mathbf {v}},E)+b({\mathbf {e}},q)=R_1({\mathbf {v}})+R_2(q), \end{aligned}$$
(243)

with

$$\begin{aligned}&R_1({\mathbf {v}}) := F({\mathbf {v}})-a({\mathbf {u}}_h,{\mathbf {v}})-b({\mathbf {v}},p_h) \\&R_2(q) := -b({\mathbf {u}}_h,q). \end{aligned}$$

Using relation (243), Young’s inequality and the properties of a and b, we can easily show that

$$\begin{aligned} \Vert E\Vert \le \frac{1}{\beta }\Vert R_1\Vert _{V'}+\frac{\nu }{\beta }\Vert \nabla {\mathbf {e}}\Vert \end{aligned}$$
(244)

and

$$\begin{aligned} \nu \Vert \nabla {\mathbf {e}}\Vert ^2 \le \frac{c_1}{\nu }\Vert R_1\Vert _{V'}^2+\frac{c_2\nu }{\beta ^2}\Vert R_2\Vert _{Q'}^2 \end{aligned}$$
(245)

with for instance \(c_1=c_2=3\), the value of these constants depending only on how we use Young’s inequality. From the last two inequalities, we deduce that the scaling (242) should be used to get

$$\begin{aligned} \nu ^k\Vert \nabla {\mathbf {e}}\Vert ^2+\nu ^{k-2}\Vert E\Vert ^2\le C\left( \nu ^{k-2}\Vert R_1\Vert _{V'}^2+\nu ^k\Vert R_2\Vert _{Q'}^2\right) , \end{aligned}$$

where C is a constant independent of \(\nu\) (but which depends on the inf-sup constant \(\beta\)).

1.2 Proof of Some Properties

We first show the relation (148) used in Sect. 4.2.2 to write the strong formulation of the problem on D. It can be proven by an integration by part back on the random domain \(D_{\omega }\) or using the Piola identity \(\nabla \cdot \left( J_{{\mathbf {x}}}A^T\right) ={\mathbf {0}}\) (see [101] for instance). Indeed, we have

$$\begin{aligned} \int _Dq|J_{{\mathbf {x}}}|(A^T\nabla )\cdot {\mathbf {v}}d{\varvec{\xi }}= & {} \int _{D_{\omega }}\tilde{q}\nabla _{{\mathbf {x}}}\cdot \tilde{{\mathbf {v}}}d{\mathbf {x}}= -\int _{D_{\omega }}\nabla _{{\mathbf {x}}}\tilde{q}\cdot \tilde{{\mathbf {v}}}d{\mathbf {x}}\\= & {} -\int _D|J_{{\mathbf {x}}}|(A^T\nabla q)\cdot {\mathbf {v}}d{\varvec{\xi }}, \end{aligned}$$

which yields (148) since \(J_{{\mathbf {x}}}\) is either positive or negative, depending if the orientation is preserved or not by the mapping. Using the second alternative, since \(\nabla \cdot (J_{{\mathbf {x}}}A{\mathbf {v}})=(\nabla \cdot (J_{{\mathbf {x}}}A^T))\cdot {\mathbf {v}}+\left( J_{{\mathbf {x}}}A^{T}\nabla \right) \cdot {\mathbf {v}}\) we have

$$\begin{aligned} \int _DqJ_{{\mathbf {x}}}(A^T\nabla )\cdot {\mathbf {v}}d{\varvec{\xi }}= & {} \int _Dq\nabla \cdot (J_{{\mathbf {x}}}A{\mathbf {v}})d{\varvec{\xi }}\\&-\int _D(\underbrace{\nabla \cdot (J_{{\mathbf {x}}}A^T)}_{={\mathbf {0}}})\cdot (q{\mathbf {v}})d{\varvec{\xi }}\\= & {} -\int _DJ_{{\mathbf {x}}}(A^T\nabla q)\cdot {\mathbf {v}}d{\varvec{\xi }}. \end{aligned}$$

We mention that the Piola identity, which is easily obtained for smooth functions, say \(C^2\) functions, is still valid (in a weak sense) for less regular functions such as \(H^1\) functions (see for instance [102, 103]).

Then, we derive the bound for the term

$$\begin{aligned} {\text {II}}_3=b({\mathbf {v}},p_{0,h};{\mathbf {y}}_0)-b({\mathbf {v}},p_{0,h};{\mathbf {y}}) \end{aligned}$$

that appear in the proof of Proposition 23. Writing \({\varvec{\xi }}=(\xi _1,\xi _2)\) and \({\mathbf {v}}=(v_1,v_2)^T\), the two terms in component form read

$$\begin{aligned} b({\mathbf {v}},p_{0,h};{\mathbf {y}}_0)= & {} -\int _D p_{0,h}\nabla \cdot {\mathbf {v}}d{\varvec{\xi }}\\= & {} -\int _D p_{0,h}\left( \frac{\partial v_1}{\partial \xi _1}+\frac{\partial v_2}{\partial \xi _2}\right) d{\varvec{\xi }}\end{aligned}$$

and

$$\begin{aligned} b({\mathbf {v}},p_{0,h};{\mathbf {y}})= & {} -\int _D p_{0,h}J_{{\mathbf {x}}}(A^T\nabla )\cdot {\mathbf {v}}d{\varvec{\xi }}\\= & {} -\int _Dp_{0,h}J_{{\mathbf {x}}}\left( A_{11}\frac{\partial v_1}{\partial \xi _1}+A_{21}\frac{\partial v_1}{\partial \xi _2}\right. \\&+\left. A_{12}\frac{\partial v_2}{\partial \xi _1}+A_{22}\frac{\partial v_2}{\partial \xi _2}\right) d{\varvec{\xi }}. \end{aligned}$$

Subtracting these two terms and using (both continuous and discrete version of) Cauchy–Schwarz’s inequality we finally obtain

$$\begin{aligned} {\text {II}}_3= & {} \int _D(J_{{\mathbf {x}}}A_{11}-1)p_{0,h}\frac{\partial v_1}{\partial \xi _1}d{\varvec{\xi }}+\int _D J_{{\mathbf {x}}}A_{21}p_{0,h}\frac{\partial v_1}{\partial \xi _2}d{\varvec{\xi }}\\&+\int _D J_{{\mathbf {x}}}A_{12}p_{0,h}\frac{\partial v_2}{\partial \xi _1}d{\varvec{\xi }}+\int _D(J_{{\mathbf {x}}}A_{22}-1)p_{0,h}\frac{\partial v_2}{\partial \xi _2}d{\varvec{\xi }}\\\le & {} \left\| (J_{{\mathbf {x}}}A_{11}-1)p_{0,h}\right\| \left\| \frac{\partial v_1}{\partial \xi _1}\right\| +\left\| J_{{\mathbf {x}}}A_{21}p_{0,h}\right\| \left\| \frac{\partial v_1}{\partial \xi _2}\right\| \\&+\left\| J_{{\mathbf {x}}}A_{12}p_{0,h}\right\| \left\| \frac{\partial v_2}{\partial \xi _1}\right\| + \left\| (J_{{\mathbf {x}}}A_{22}-1)p_{0,h}\right\| \left\| \frac{\partial v_2}{\partial \xi _2}\right\| \\\le & {} \left( \left\| (J_{{\mathbf {x}}}A_{11}-1)p_{0,h}\right\| ^2+\left\| J_{{\mathbf {x}}}A_{21}p_{0,h}\right\| ^2+\left\| J_{{\mathbf {x}}}A_{12}p_{0,h}\right\| ^2\right. \\&\left. +\left\| (J_{{\mathbf {x}}}A_{22}-1)p_{0,h}\right\| ^2\right) ^{\frac{1}{2}}\left( \sum _{i,j=1}^2\left\| \frac{\partial v_i}{\partial \xi _j}\right\| \right) ^{\frac{1}{2}} \\= & {} \left\| (J_{{\mathbf {x}}}A^T-I)p_{0,h}\right\| \left\| \nabla {\mathbf {v}}\right\| . \end{aligned}$$

To conclude, we give the proof of Proposition 22, which is inspired by what is done in [104] for the deterministic steady Navier–Stokes equations.

Proof

In what follows, all equations depending on \({\mathbf {y}}\) hold \(\rho\)-a.e. in \(\varGamma\), without specifically mentioning it. Moreover, the dependence of the functions with respect to \({\mathbf {y}}\in \varGamma\) will not necessarily be indicated. Let \({\mathbf {e}}({\mathbf {y}}):={\mathbf {u}}({\mathbf {y}})-{\mathbf {u}}_{0,h}\) and \(E({\mathbf {y}}):=p({\mathbf {y}})-p_{0,h}\). Then (162) yields

$$\begin{aligned}&a({\mathbf {e}},{\mathbf {v}};{\mathbf {y}})+b({\mathbf {v}},E;{\mathbf {y}})+b({\mathbf {e}},q;{\mathbf {y}}) \nonumber \\&\quad +D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}})=R(({\mathbf {v}},q);{\mathbf {y}}) \end{aligned}$$
(246)

for all \(({\mathbf {v}},q)\in V\times Q\), where

$$\begin{aligned} D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}}):=c({\mathbf {u}},{\mathbf {u}},{\mathbf {v}};{\mathbf {y}})-c({\mathbf {u}}_{0,h},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}}). \end{aligned}$$

We can show that

$$\begin{aligned} D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}})\le (2\theta \nu \alpha +\hat{C}\Vert \nabla {\mathbf {e}}_0\Vert )\Vert \nabla {\mathbf {e}}\Vert \Vert \nabla {\mathbf {v}}\Vert \end{aligned}$$
(247)

and

$$\begin{aligned} D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {u}}-{\mathbf {u}}_{0,h};{\mathbf {y}})\le (\theta \nu \alpha +\hat{C}\Vert \nabla {\mathbf {e}}_0\Vert )\Vert \nabla {\mathbf {e}}\Vert ^2 \end{aligned}$$
(248)

where \({\mathbf {e}}_0:={\mathbf {u}}_{0}-{\mathbf {u}}_{0,h}\) and M, \(\alpha\) and \(\hat{C}\) are defined in Proposition 19. Indeed, for any \({\mathbf {v}}\in V\) we have

$$\begin{aligned} D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}})= & \, c({\mathbf {u}},{\mathbf {u}}-{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}}) \\&+\,c({\mathbf {u}}-{\mathbf {u}}_{0,h},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}}) \\\le & {} \hat{C}\left( \Vert \nabla {\mathbf {u}}\Vert +\Vert \nabla {\mathbf {u}}_0\Vert +\Vert \nabla {\mathbf {e}}_0\Vert \right) \Vert \nabla {\mathbf {e}}\Vert \Vert \nabla {\mathbf {v}}\Vert \\\le & {} \hat{C}\left( 2\theta \frac{\alpha \nu }{\hat{C}}+\Vert \nabla {\mathbf {e}}_0\Vert \right) \Vert \nabla {\mathbf {e}}\Vert \Vert \nabla {\mathbf {v}}\Vert \end{aligned}$$

thanks to (156), which proves relation (247). For (248), we proceed analogously using the fact that \(c({\mathbf {u}},{\mathbf {v}},{\mathbf {v}};{\mathbf {y}})=0\) for any \({\mathbf {v}}\in V\). The rest of the proof consists of two steps, first the derivation of a bound on \(\Vert E\Vert\) and then a bound on \(\Vert \nabla {\mathbf {e}}\Vert\).

Using the inf-sup condition (153) for b, the bound (247) on D, the continuity of a and the relation (246) with \(q=0\), we have

$$\begin{aligned} \Vert E\Vert\le & {} \frac{1}{\beta }\sup _{{\mathbf {v}}\in V}\frac{|b({\mathbf {v}},p-p_{0,h};{\mathbf {y}})|}{\Vert \nabla {\mathbf {v}}\Vert } \nonumber \\= & {} \frac{1}{\beta }\sup _{{\mathbf {v}}\in V}\frac{|R_1({\mathbf {v}};{\mathbf {y}})-a({\mathbf {u}}-{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}})-D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {v}};{\mathbf {y}})|}{\Vert \nabla {\mathbf {v}}\Vert } \nonumber \\\le & {} \frac{1}{\beta }\left[ \Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}+(\nu M+2\nu \alpha +\hat{C}\Vert \nabla {\mathbf {e}}_0\Vert )\Vert \nabla {\mathbf {e}}\Vert \right] . \end{aligned}$$
(249)

Therefore, using the relation \((a+b)^2\le 2(a^2+b^2)\) we obtain

$$\begin{aligned} \frac{1}{\nu }\Vert E\Vert ^2\le & {} \frac{2}{\beta ^2\nu }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}^2 \nonumber \\&+\frac{2(M+2\alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert )^2}{\beta ^2}\nu \Vert \nabla {\mathbf {e}}\Vert ^2. \end{aligned}$$
(250)

We now give a bound on the error \(\Vert \nabla {\mathbf {e}}\Vert\) for the velocity. Using the inequalities (248) and (249), the coercivity of the bilinear form a, Young’s inequality several times and taking \({\mathbf {v}}={\mathbf {e}}\) and \(q=-E\) in (246), we get

$$\begin{aligned} \nu \alpha \Vert \nabla {\mathbf {e}}\Vert ^2\le & {}\, a({\mathbf {e}},{\mathbf {e}};{\mathbf {y}}) \nonumber \\= & {}\, R_1({\mathbf {e}};{\mathbf {y}})-R_2(E;{\mathbf {y}})-D({\mathbf {u}},{\mathbf {u}}_{0,h},{\mathbf {e}}) \nonumber \\\le & {}\, \Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}\Vert \nabla {\mathbf {e}}\Vert +\Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'}\Vert E\Vert \nonumber \\&\,+(\theta \nu \alpha+\hat{C}\Vert \nabla {\mathbf {e}}_0\Vert )\Vert \nabla {\mathbf {e}}\Vert ^2 \nonumber \\\le & {} \frac{1}{2\gamma _1\nu }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}^2+\frac{\nu }{2\beta ^2\gamma _2}\Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'}^2 \nonumber \\&+\frac{1}{\beta }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}\Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'} \nonumber \\&+\left( \frac{\gamma _1}{2}+\frac{\gamma _2(M+2\alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert )^2}{2}\right. \nonumber \\&\left. +\,\theta \alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert \right) \nu \Vert \nabla {\mathbf {e}}\Vert ^2 \nonumber \\\le & {} \frac{c_1}{\nu }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}^2+c_2\nu \Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'}^2 \nonumber \\&+\,\left( \frac{\gamma _1}{2}+\frac{\gamma _2(M+2\alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert )^2}{2}\right. \nonumber \\&\left. +\theta \alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert \right) \nu \Vert \nabla {\mathbf {e}}\Vert ^2, \end{aligned}$$
(251)

with

$$\begin{aligned} c_1 = \frac{1}{2\gamma _1}+\frac{1}{2} \quad \text{ and } \quad c_2 = \frac{1}{2\gamma _2\beta ^2}+\frac{1}{2\beta ^2}. \end{aligned}$$

Recalling that \(\theta \in [0,1[\) and using the convergence of \({\mathbf {u}}_{0,h}\) to \({\mathbf {u}}_0\) as h tends to 0, we can choose h, \(\gamma _1\) and \(\gamma _2\) small enough so that

$$\begin{aligned} \frac{\gamma _1}{2}+\frac{\gamma _2(M+2\alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert )^2}{2}+\theta \alpha +\frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert \le \frac{1+\theta }{2}\alpha . \end{aligned}$$
(252)

For instance, we can choose h small enough so that

$$\begin{aligned} \frac{\hat{C}}{\nu }\Vert \nabla {\mathbf {e}}_0\Vert \le \frac{1-\theta }{6}\alpha \end{aligned}$$
(253)

and take

$$\begin{aligned} \gamma _1=\frac{1-\theta }{3}\alpha \quad \text{ and } \quad \gamma _2=\frac{1-\theta }{3(M+2\alpha +\frac{1-\theta }{6}\alpha )^2}\alpha \end{aligned}$$

which depends only on \(\theta\), \(\sigma _{{{ min}}}\) and \(\sigma _{{{ max}}}\). Therefore, the last term of the right-hand side of inequality (251) can be moved to the left and we get

$$\begin{aligned} \nu \Vert \nabla {\mathbf {e}}\Vert ^2\le \frac{2}{(1-\theta )\alpha }\left[ \frac{c_1}{\nu }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}^2+c_2\nu \Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'}^2\right] . \end{aligned}$$
(254)

Using this bound in (250) together with (253) we get

$$\begin{aligned} \frac{1}{\nu }\Vert E\Vert ^2\le & {} \left( \frac{2}{\beta ^2}+\frac{4c_1}{3\gamma _2\beta ^2}\right) \frac{1}{\nu }\Vert R_1(\cdot ;{\mathbf {y}})\Vert _{V'}^2 \\&+\frac{4c_2}{3\gamma _2\beta ^2}\nu \Vert R_2(\cdot ;{\mathbf {y}})\Vert _{Q'}^2. \end{aligned}$$

Replacing finally \({\mathbf {y}}\) by \({\mathbf {Y}}(\omega )\), the combination of last two inequalities permits to conclude the proof since \(c_1\) and \(c_2\) depend only on \(\beta\) as well as \(\gamma _1\) and \(\gamma _2\), which in turn depend only on \(\theta\), \(\sigma _{{{ min}}}\) et \(\sigma _{{{ max}}}\). \(\square\)

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Guignard, D. Partial Differential Equations with Random Input Data: A Perturbation Approach. Arch Computat Methods Eng 26, 1313–1377 (2019). https://doi.org/10.1007/s11831-018-9275-2

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